Edelstein-Keshet L. Mathematical Models in Biology
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366
Continuous Processes
and
Ordinary
Differential
Equations
16.
Morris
and
Lecar (1981) describe
a
semiqualitative model
for
voltage
oscilla-
tions in the
giant muscle
fiber of the
barnacle.
In
this system
the
important ions
are
potassium
and
calcium (not sodium).
The
equations they suggest
are the
following:
where
v
=
voltage,
M
=
fraction
of
open Ca
2+
channels,
N
=
fraction
of
open
K
+
channels.
(a)
Interpret these three equations.
For
reasons detailed
in
their papers Mor-
ris and
Lecar
define
the
functions
M«,,
A
w
, M», and A# as
follows:
(b)
Sketch
or
describe
the
voltage dependence
of
these
functions.
(c)
Morris
and
Lecar
consider
the
reduced
vN
system
to be an
approximation
to the
whole model. What assumption underlies this approximation?
(d)
Show that
in the
reduced
vN
system
the
variables
are
constrained
to
sat-
isfy
the
following inequalities:
(You
may
need
to use the
fact
that
0 < M < 1.)
(e)
Give support
for the
configuration
of
nullclines shown
in
Figure
8.
\l(d).
Notice
mat
there
is an
intersection within
the
region described
by the in-
equalities
in
part
(d) of
this problem.
(f)
Suppose
you are
told that
at the
steady state
the
linearized system
has a
pair
of
complex eigenvalues
A = a + bi
such that
a + bi
behaves
as in
Figure
8.
\l(b)
as the
current
/
changes.
Use
your results
from
parts
(a) to
(e)
to
make
a
statement about
the
existence
and
stability
of a
limit-cycle
solution.
Limit
Cycles,
Oscillations,
and
Excitable
Systems
367
(g)
Interpret this
in the
biological context
of
electrical signal propagation
in
the
barnacle giant muscle
fiber,
(h)
Compare
the
assumptions
and
results obtained
from
this model
to the
Hodgkin-Huxley
and
Fitzhugh models.
(This problem could
be
extended
to an
independent project with in-class dis-
cussion.
See
Morris
and
Lecar, 1981,
for
other
details.)
17.
Limit cycles
in
predator-prey
systems.
In
this problem
we
investigate details
that
arise
in
Section 8.7.
(a)
Justify
the
particular
form
of
equations (42a,b) used
in
discussing
a
predator-prey system.
*(b) Suppose
the
orientations
of the
nullclines
is as
shown
in
Figure 8.20(a).
Use
conditions
1 and 2 in
Section 8.7, along with
the
fact
that
on
these
curves/(jc,
v) = 0 and
g(x,
y) = 0
respectively,
to
reason that
the
direc-
tion
of flow
along nullclines must conform
to
that shown
in
Figure
8.20(a).
(Hint:
Use the
inequalities
to
determine
in
which
regions/,
or g,
must
be
positive
and in
which negative.)
(c)
A
condition
for a
limit cycle
is
that
x
2
> x\.
Interpret
this
biologically
and
describe
why
this inequality
is
necessary.
(d)
Verify
condition
7
[equation (48)]
by
implicit
differentiation
of
g(x,y)
= Q.
(e) A
steady state
(x, y) is
unstable
if any one of the
eigenvalues
of the
equa-
tions
[linearized
about
(x, y)] has a
positive real part.
Why
then
is it
nec-
essary
to
have
two
positive eigenvalues
to
ensure that
a
limit cycle exists?
18. In
this problem
we
compute
the
stability properties
of the
nonzero steady state
of
the
predator-prey equations (42a,b).
(a)
Find
the
Jacobian
of
equations (42a,b).
(b)
Show that conditions
for an
unstable node
or
spiral
at (x, y) are as
given
in
equations
(5la-b).
(c)
Verify
the
following relationship between
the
slope
5^ of the
nullcline
g
= 0 and the
partial derivatives
of g:
(d) Use the
inequality
a + d > 0 and the
inequalities
f
y
< 0, g
x
> 0, and
g
y
< 0 to
establish
the
following
result:
for (x, y) to be an
unstable node
or
spiral,
it is
necessary
and
sufficient
that
Why
does this imply that
Why
does
it
follow that
in
other words, that
the
slope
of the
curve f(x,
y) = 0
must
be
positive
at
the
steadv state?
368
Continuous Processes
and
Ordinary
Differential
Equations
(e) Use the
inequality
ad — be > 0 to
show that
at the
steady state
Note:
all
quantities
are
evaluated
at the
steady state,
as in
part (i). Reason
that
this inequality implies
that
19. The
following predator-prey system
is
discussed
by May
(1974):
(a)
Interpret
the
terms appearing
in
these equations
and
suggest what
the
var-
ious
parameters might represent.
(b)
Sketch
the H and P
nullclines
on an HP
phase plane.
(c)
Apply Bendixson's criterion
and
Dulac's criterion (for
B = 1
/HP).
20. A
number
of
modifications
of the
Lotka-Volterra predator-prey model that
have been suggested over
the
years
are
given
in the
last
box in
Section 6.2.
By
considering several combinations
of
prey density-dependent growth
and
preda-
tor
density-dependent attack rate, determine whether such modifications might
lead
to
limit-cycle oscillations.
You may
wish
to do the
following:
(a)
Check
to see
whether
the
Kolmogorov conditions
are
satisfied.
(b)
Plot nullclines
by
hand
(or by
writing
a
simple computer program
and
check
to see
whether
the
Poincare-Bendixson conditions
are
satisfied.
(c)
Determine
the
stability properties
of
steady states.
21.
This problem
arises
in a
model
of a
plant-herbivore system: (Edelstein-Keshet,
1986).
Assume that
a
population
of
herbivores
of
density
y
causes changes
in
the
vegetation
on
which
it
preys.
An
internal variable
x
reflects
some physical
or
chemical property
of the
plants which undergo changes
in
response
to
her-
bivory.
We
refer
to
this attribute
as the
plant
quality
of the
vegetation
and as-
sume that
it may in
turn
affect
the fitness or
survivorship
of the
herbivores.
If
this happens
in a
graded,
continuous interaction, plant quality
may be
modeled
by
a
pair
of
ODEs such
as
Sketch this
as a
function
or x and
reason that
the
plant quality
x
always
remains within
the
interval
(0, 1) if
*(0)
is in
this range. Show that plant
quality
may
either
decrease
or
increase depending
on (1)
initial value
of x
(a) In one
case
the
function/(jc,
y) is
assumed
to be
Limit
Cycles,
Oscillations,
and
Excitable
Systems
369
and
(2)
population
of
herbivores.
For a
given herbivore population den-
sity
y,
what
is the
"breakeven" point (the level
of x for
which
dx/dt
= 0)?
(b) It is
assumed that
the
herbivore population undergoes logistic growth (see
Section
6.1) with
a
carrying capacity that
is
directly proportional
to
cur-
rent
plant quality
and
reproductive rate
/3.
What
is the
function
g?
(c)
With
a
suitable definition
of
constants
in
this problem your equations
should have
the
following
nullclines:
Draw
these curves
in the xy
plane. (There
is
more
than
one
possible
configuration,
depending
on the
parameter values.)
(d) Now find the
direction
of
motion along
all
nullclines
in
part (c). Show
that
under
a
particular configuration there
is a set in the xy
plane that
"traps"
trajectories.
(e)
Define
y =
a/(aK
- 1).
Interpret
the
meaning
of
this parameter. Show
that
(y, Ky) is a
steady state
of
your equations
and
locate
it on
your phase
plot.
Find
the
other steady state.
(f)
Using stability analysis, show that
for y > 1, (y, Ky) is a
saddle point
whereas
for y < 1 it is a
focus.
(g) Now
show that
as j3
decreases
from
large
to
small values,
the
steady state
(y
< 1)
undergoes
the
transition
from a
stable
to an
unstable
focus.
(h) Use
your
results
in the
preceding parts
to
comment
on the
existence
of
periodic solutions. What would
be the
biological interpretation
of
your
answer?
22.
Lotka's
chemical model given
in
Section
8.8 is
dynamically equivalent
to the
Lotka-Volterra predator-prey model. Thus modifications
in the
chemical kinet-
ics
should result
in
system(s) that exhibit
limit-cycle
oscillations.
Explore
whether
the
Kolmogorov conditions (see
box in
Section 8.8) apply
to a
chemi-
cal
system;
if so,
interpret
the
inequalities
(points
1
through
8 in the
box)
in the
context
of
chemical (rather than animal)
species.
23.
Exact equations
and
Lotka's model.
Let M and N be the
functions
defined
by
equations (60),
and let V be
some
function
satisfying
Define
370
Continuous Processes
and
Ordinary
Differential
Equations
The
integrals
are to be
performed with respect
to one of the
variables only,
the
other
one
being held
fixed.
Show that
K2
If
V\ = V
2
= V,
show that
V
must
be
given
by
equation (55).
24. The
following equations were developed
by
Goodwin (1963)
as a
model
of
protein-mRNA
interactions:
Show that this system
is
conservative
and has
oscillatory solutions.
In
the
remaining problems
we
consider systems
in
which
it is
possible
to
explicitly
solve
for
limit cycles
and
deduce their stability
by
transforming variables
to
polar
form.
25.
Consider
a
system
of
equations such
as
(la,b). Transform variables
by
defining
(b)
Conclude that there
are
circular solutions. What
is the
direction
of
rota-
tion?
Are
these cycles stable?
27.
Show that
the
system
(b) By
transforming variables, obtain
26.
Consider
the
equations
(c)
Verify
that
(b)
Show that
(a)
Show that
Limit
Cycles,
Oscillations,
and
Excitable
Systems
371
has
closed
elliptical orbits.
(Hint:
Consider
first
transforming
xtox
- 5 and y
to y/2 and
then using
a
polar transformation.)
28.
Consider
the
nonlinear system
of
equations
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Circadian
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as
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The
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2d ed.
Springer-Verlag,
New
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L.N.,
ed.
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C. A.;
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S. F.; and
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Limit
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375
APPENDIX
1 TO
CHAPTER
8:
SOME BASIC TOPOLOGICAL NOTIONS
Many
of the
theorems that apply
to
systems
of two
ordinary
differential
equations
such
as
(la,b)
are
based
on
what
are
called
topological
properties
of
curves
in the
plane. While
the
topological theory itself
is
abstract
and
exceptionally beautiful,
we
shall avoid formal details,
giving
instead
a
descriptive outline
of
some
key
concepts.
(See
Arnold,
1981,
for an in-
troductory summary.)
Imagine
a
vector
field in the xy
plane.
The
vector
field may
have been generated
by the
set of
equations (la,b). Figure
8.24
serves
as an
example.
In
this vector
field we
place
a
sim-
ple
oriented closed curve;
(we may
think
of
this curve
as a
deformed circle along which some
point
moves
in a
particular direction).
We first
consider arbitrary curves
as
"test"
objects that
are
used
to
understand
the
vector
field.
(Only later will
we
turn
to
closed
curves generated
by
the
vector
field
itself.) Tracing
the
path
of a
single point around
the
test curve
we may
follow
the
rotation
of a field
vector attached
to the
point.
If the
curve does
not go
through
any
singu-
lar
point
[point
at
which
F = G = 0 in
equations
(1)],
the
rotation
will
be
continuous
as the
curve
is
traversed,
and the field
vector
will
have returned
to its
original orientation
when
the
point
has
returned
to its
initial position.
The
number
of
revolutions
and the
direction
of
rota-
tion
executed
by the field
vector depend
on
details
of the flow
patterns. (Several possibilities
are
shown
in
Figure
8.24.)
The
index
of
a
curve
is the
number
of
revolutions,
and the
sign
of
the
index reflects whether
the
rotation
is in the
same
sense
or in an
opposite
sense
to
that
of
the
curve.
A
number
of
properties make
the
concept
of the
index important:
1. The
index does
not
change
if the
curve
is
distorted, twisted,
or
enlarged, provided that
in
the
process
of
change
the
curve does
not go
through
any
singular points.
2.
Similarly,
if the
vector
field is
warped, distorted,
or
rearranged,
the
index
of the
curve
will
not
change
if no
singular points cross
the
curve
or are on the
curve during
the
process.
Using such properties
it can be
shown
(see
Problem
3la)
that
the
following
fact
holds
true:
3. The
index
of a
simple closed curve
is
zero unless there
is at
least
one
singular
point
in
the
region bounded
by the
curve.
We now
define
the
index
of
a
singular
point
P as the
index
of any
circle
of
small radius
centered
at P.
(See
Figure 8.24.) According
to
property
1 the
radius
of
this circle
is
immaterial
as
long
as the
curve
encloses
only
one
singular point
and
does
not
itself
go
through
any
singular points.
4. The
indices
of
common isolated singular points
are as
follows
[see
Figure
8.24(/-/i)]:
(i) The
index
of a
node
is +1.
(ii)
The
index
of a
focus
is
-I-1.
(iii)
The
index
of a
center
is +1.
(iv)
The
index
of a
saddle point
is -1.
(These
are
independent
of
stability
in
cases
1 and 2 and
independent
of
direction
of
rotation
in
cases
3 and 4. See
problem 31b.)